### YOUR CODE HERE ###
def get_sorted_singular_values_and_vectors(data_std):
# Perform Singular Value Decomposition
U, singular_values, Vt = np.linalg.svd(data_std)
# U contains left singular vectors (not needed for SVD-based PCA)
# singular_values contains the singular values
# Vt contains the right singular vectors, which are the principal components
return singular_values, Vt
def plot_city_figures_SVD(original_df, city_name):
# Get Original Data
city_data = original_df.loc[city_name]
dates = original_df.columns #for plotting later
scaler = StandardScaler()
data_raw_std = scaler.fit_transform(original_df)
# Get standardized city data
city_data_std = scaler.transform(original_df.loc[city_name].values.reshape(1, -1))
# Calculate indices for 8 evenly spaced dates
spaced_indices = np.linspace(0, len(dates) - 1, 8, dtype=int)
spaced_dates = dates[spaced_indices]
# Plot Original Times Series:
plt.figure(figsize=(10, 4))
plt.plot(dates, city_data, label="Original Data", linewidth=2)
plt.xlabel("Dates", fontsize=14)
plt.ylabel("Temperature", fontsize=14)
plt.title(f"Original Data: {city_name}", fontsize=16)
plt.xticks(spaced_dates)
plt.show()
# Get Singular Values and Vectors:
singular_values, Vt = get_sorted_singular_values_and_vectors(data_raw_std)
# Incremental Reconstruction using SVD
SVs = [0, 1, 2, 5, 15] # Principal components to be used for reconstruction
# Updated loop for incremental reconstruction using SVD
plt.figure(figsize=(10, 5))
for i in range(5):
# Get first x singular vectors
V = Vt[:SVs[i], :]
# Reconstruct the data from the singular vectors by projection
reconstructed_data = np.dot(np.dot(city_data_std, V.T), V)
# De-standardize the data
reconstructed_data_nonstandard = scaler.inverse_transform(reconstructed_data).flatten()
# Plot
plt.plot(dates, reconstructed_data_nonstandard, label=f"SVs: {SVs[i] + 1}")
plt.plot(dates, city_data, label="Original Data", linewidth=2)
plt.xlabel("Dates", fontsize=14)
plt.ylabel("Temperature", fontsize=14)
plt.title(f"Reconstruction with Incremental Singular Values: {city_name}", fontsize=16)
plt.xticks(spaced_dates)
plt.legend()
plt.tight_layout()
plt.show()
#calculate residuals
residuals = reconstructed_data_nonstandard.flatten() - city_data.values
plt.figure(figsize=(10, 4))
plt.plot(dates, residuals)
plt.xlabel("Dates", fontsize = 14)
plt.ylabel("Residuals (original - reconstruction)", fontsize = 14)
plt.title(f"Final Reconstruction Residuals: {city_name}", fontsize = 16)
plt.xticks(spaced_dates)
plt.show()
#Get RMSE Plot by doing a seperate loop:
num_comps = 10
RMSE = np.zeros(num_comps)
for i in range(num_comps):
# Get first x singular vectors
V = Vt[:i, :]
# Reconstruct the data from the singular vectors
reconstructed_data = np.dot(np.dot(city_data_std, V.T), V)
# De-standardize the data
reconstructed_data_nonstandard = scaler.inverse_transform(reconstructed_data).flatten()
#calculate residuals
residuals = reconstructed_data_nonstandard.flatten() - city_data.values
#calculate RMSE
RMSE[i] = np.sqrt(np.mean(residuals**2))
plt.figure(figsize=(10, 4))
plt.plot(np.arange(1,num_comps+1), RMSE)
plt.xlabel("Number of Components", fontsize = 14)
plt.ylabel("Root Mean Squarred Error", fontsize = 14)
plt.title(f"RMSE of reconstruction: {city_name}", fontsize = 16)
plt.show()
return RMSE, reconstructed_data_nonstandard
cities = ['Yakeshi', 'Zamboanga','Norilsk', 'Juliaca', 'Doha']
for city in cities:
plot_city_figures_SVD(data_raw, city)
